Localised Davies generators for (pseudo)differential operators

Abstract

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyén proposed a construction involving localisation in time and carried out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad evolution is expected to settle to the Gibbs state. In this paper, we show that the localised Davies construction also works for natural classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution. Our emphasis is microlocal: we prove that the localised jump operators are themselves pseudodifferential, and hence pseudolocal. The proof involves a novel version of Egorov's theorem valid for all times.

A slightly modified title and abstract; a new introduction; an expository appendix on an explicit model added; misprints and small mistakes corrected